Date: Mar 24, 2013 8:57 PM
Author: ross.finlayson@gmail.com
Subject: Re: Matheology § 224
On Mar 24, 4:48 pm, "Ross A. Finlayson" <ross.finlay...@gmail.com>wrote:> On Mar 24, 4:16 pm, Virgil <vir...@ligriv.com> wrote:>>>>>>>>>> > In article> > <ab63154d-ab7e-4a03-be1b-17ac93226...@hd10g2000pbc.googlegroups.com>,> > "Ross A. Finlayson" <ross.finlay...@gmail.com> wrote:>> > > On Mar 24, 2:51 pm, fom <fomJ...@nyms.net> wrote:> > > > On 3/24/2013 4:34 PM, WM wrote:>> > > > > On 24 Mrz., 21:29, Virgil <vir...@ligriv.com> wrote:> > > > >> In article> > > > >> <729f073f-8948-4eb9-991a-2bd249ac5...@c6g2000yqh.googlegroups.com>,>> > > > >> A binary tree that contains one path of each positive natural number> > > > >> length will necessarily also contain exactly one path of infinite length.>> > > > > Like the sequence> > > > > 0.1> > > > > 0.11> > > > > 0.111> > > > > ...> > > > > that necessarily also contains its limit?>> > > A binary tree that contains one path, of all zero-branches, of each> > > finite length, will necessarily contain a path of 0-branches of> > > infinite length.>> > Better is>> > root> > | \> > 0 1> > | \> > 0 1> > | \> > 0 1> > | \> > 0 1> > | \> > 0 1> > | \> > 0 1> > | \> > 0 1> > | \>> > And so on>> > -->> A binary tree, of all 0's or 1's (0- or 1- branches) with paths of> each finite length, has a path of infinite length: that is a subtree> of the intersection[*], of the finite paths. A tree with paths of> each length n having 0's except the last 1 shows a binary tree with an> infinite path: that is a subtree of the union of the finite paths.> As a space-complexity problem it's simpler to maintain the> intersection[*] than the union.>> * [ To be sure, to well-define intersection (i.e. to keep it true), it> is as the union where the existence of a disjoint leaves the result> undefined. Basically this of the difference among unary and> infinitary, and binary and n-ary.]>> Basically in entropy coding and Huffman coding there are codes with> the prefix property that codes of arbitrary width can be put into a> sequence and demarcated courtesy the prefix property. Then, an idea> is that with paths as codes with the prefix property and distinct> length, there would be in the tree a path of each finite length, but> that paths of greater length would have a different prefix than each> preceding path.>> https://en.wikipedia.org/wiki/Prefix_codehttps://en.wikipedia.org/wiki/Huffman_coding>> Then, where Huffman coding is to find the largest alphabet with the> prefix property with the least average length of the path, here the> idea for dense bounded tree coding is to find an alphabet with the> prefix property with one code for each length: with the maximum suffix> of the path that is disjoint the other code's paths, that each pair is> mutually disjoint.>> Basically this is about codes with the prefix property (that there> would be a branch from lesser length codes, and the next), the> palindrome property, or otherwise with the prefix property, and that> the suffix was not following the suffix of another code initial> segment, in reverse as it were. Basically it is that the reverse of> the code of length n as path, would have the prefix property, to the> initial segment of length n of any code of greater length.>> Then, saying that there would always be an infinite path in the union> of paths of each finite length of the binary tree is that there is no> "dense bounded" coding as here. Basically that is a question of what> are the most paths and most lengths (and as to complexity and entropy)> there can be with the least amount of infinite paths.>> Here, again, the intersection[*] of the 0-paths is infinite, the union> of the 0*1-paths has a subtree that is infinite, a distinct> difference: the least number of finite paths, with an infinite path.>> There is a difference among representations, of unary, infinitary,> binary, and n-ary (in spaces), structural difference with> concomitantly various results, as of their variety.>

This as well get directly to questions on sampling the real numbersvia Bernoulli trials. Flip a coin, it begins a sample, flip another,it begins another, and refines the first, and so on.

The idea here is to maximize the unique part of the path, sharing theleast among their union. Obviously with 2 paths and maximum variationeach of the first level of the tree is exhausted, with 2 levels thereare 4 possible paths each having their own, with n levels (of thebalanced tree) there are 2'n distinct paths of length n, that sharewith (2^n-m)-1 other paths their initial segments length m. Thebalanced tree has the most variation, the single path the least. Thenthe question of there being a "dense bounded" coding (of natural n>0to path of length n) is as to there being enough paths through the b-ary balanced tree, with above that b = 2, that each path is mutuallydisjoint. There is an obvious case for b = l that there exist pathsup to length l that would not share initial segments and theirintersection[*] would be empty thus finite, and their union would befinite. Then, in n-ary, up to n-many paths of distinct lengths up tol, could be mutually disjoint. For length n+1, at least two pathsshare an initial segment of length 1, for n + k<n, the sum of thelength of the shared initial segment goes to n, or here there is afilling in terms of maximizing variety.

So, there are b^l or b^p (l defined to be p for precision) many pathsin the b-ary balanced tree of depth p. For one path of each length,the ratio of paths in the path-length-replete tree to the balancedtree is p / b^p. As the base of the tree increases, there are manymore paths to be distinct for more of their branches, that it takesb^p branches to saturate each level of depth of the tree, thus tolength b^p.

In the asymptotic: b^p >> p. A tree with only a path for eachlength , grows sparse in the union of the paths, where not socontrived, eg branching at random.

Basically then for a tree to have no dense coding, for a n-lengthedcoding of n E N, with a maximally dispersed coding, will have that acode of length p, will be the initial segment, of a code of lengthb^p.

p -> b^p -> b^(b^p) -> b^(b^(b^p) -> ...

Hmm, that doesn't very simply lend itself to notation as tetration,the sequence of lengths of paths that would form an infinite path inthe b-ary tree with maximally dispersed paths where no "dense bounded"coding exists.